Uh, really?
Well, it could be passed off as a narration quirk, after all, Shirou is technically from the future and know her as a baroness.
(copied from ukpge Can you stand for Election)
Disqualifying offices
Certain post-holders are disqualified from becoming a 1.5Member of Parliament. These include:
• civil servants
• members of police forces
• members of the armed forces
• government-nominated directors of commercial companies
• judges
• members of the legislature of any country or territory outside the Commonwealth
•
peers who sit and can vote in the House of Lords
• bishops of the Church of England (known as the Lords Spiritual) who are entitled to sit and vote in the House of Lords
Technically a Prime Minister can be a member of the house of lords but this hasn't happened for a long time and they wouldn't have a position in the commons
Having a Life Peerage means you can sit and vote in the house of lords
Life peer - Wikipedia (mentions when she got her peerage)
It isn't rare that Prime Ministers will get a Life Peerage when they leave office (although not all get the offer and some refuse)
Basically, one of the major aspects of our civil war was to prevent the royal family and members of the house of lords from overly influencing the house of commons... basically the house of common's job is to represent the common people
There are ways to resign from titles but that is rare.